Finite CKS sparse QLSP with variable-time inverse estimation

Inputs. A finite Hermitian $d$-sparse $A$ with $\mathrm{\parallel }A{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le 1$, normalized $\mathrm{\mid }b⟩$ supported on eigenvalues $\mathrm{\mid }\lambda \mathrm{\mid }\ge 1\mathrm{/}\kappa$, rational $0<e<1$ and rational failure budget $0<\rho <1\mathrm{/}3$, the full reversible row/reverse-location and entry-value access of Childs–Kothari–Somma sparse QLSP oracle model, and supplied $B,B^{†}$ preparing $\mathrm{\mid }b⟩$. The preparation contract is $B(\mathrm{\mid }0{⟩}_I\mathrm{\mid }0{⟩}_{W_B})=\mathrm{\mid }b{⟩}_I\mathrm{\mid }0{⟩}_{W_B}$ for a declared preparation-work register $W_B$; stage lists act as identity on $W_B$. Every controlled, inverse, state-preparation, simulation, reflection, arithmetic, classical table, precision, and compilation cost is an input coordinate.

Modes. The input selects efficient-qsvt or the original fourier-table inverse constructor, and retains its declared capped or uncapped construction policy. The Fourier table, coefficient-generation and common-normalization instructions below are the complete original alternate mode. Their generic table/preparation costs remain charged. The complexity guarantee uses efficient-qsvt; no such guarantee is inferred for an arbitrary supplied Fourier SELECT circuit or generic coefficient table.

Finite-precision matrix and simulation budget (both modes). Let $\tilde{A}$ be the Hermitian sparse matrix decoded from the finite-precision value oracle, and require a certified induced error $\mathrm{\parallel }A-\tilde{A}{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le {ϵ}_{\mathrm{v}\mathrm{a}\mathrm{l}}$; exact value access is the case ${ϵ}_{\mathrm{v}\mathrm{a}\mathrm{l}}=0$. Require ${\tilde{A}}_{ij}=0$ whenever $(i,j)$ is absent from the supplied location lists; the lists must cover every nonzero decoded entry. A decoded entry outside those lists, a non-Hermitian decoded table, an entry with $\mathrm{\mid }{\tilde{A}}_{ij}\mathrm{\mid }>1$, or a missing error certificate returns INVALID_ACCESS. Construct the exact sparse block encoding with compression $\tilde{A}\mathrm{/}d$ and normalization $d$ by the QA25 supplied-permutation $U_L^{†}{U}_R$ circuit using the known entry bound $h=1$ (not the unknown maximum entry). Use the Hermitian row-location and reverse-location permutations on swapped system registers to supply the column enumerations; charge the swaps and every inverse/control form. The value flag circuit is: On a valid value slot use the flag rotation $\left(\begin{array}{cc}{\tilde{A}}_{ij}& -\sqrt{1-\mathrm{\mid }{\tilde{A}}_{ij}{\mathrm{\mid }}^2}\\ \sqrt{1-\mathrm{\mid }{\tilde{A}}_{ij}{\mathrm{\mid }}^2}& \overline{{\stackrel{~}{A}}_{ij}}\end{array}\right)$, with value query/unquery as in that circuit. This changes only the value amplitude in the QA25 supplied-permutation circuit, so its row/reverse-location, value-query, uniform-state-preparation, inverse, and controlled-call counts are unchanged and separately charged. The location maps enumerate the support of $A$ and may therefore list additional zero-valued entries of $\tilde{A}$; their flag-zero contribution is zero. In efficient-qsvt mode, for each QSVT stage, ${ϵ}_{\mathrm{B}\mathrm{E},j}$ is a certified bound on its actual PUE compression error relative to $A\mathrm{/}d$, including at least ${ϵ}_{\mathrm{v}\mathrm{a}\mathrm{l}}\mathrm{/}d$ and any block-circuit error; the stage is admissible only if $4D_j\sqrt{{ϵ}_{\mathrm{B}\mathrm{E},j}}\le {\eta }_I\mathrm{/}8$.

Let $\mathcal{Q}$ be the finite multiset of simulator-call sites in one complete $A$ list and one complete $A^{\mathrm{\prime }}$ list, including every dyadic QPE power and coherent repetition. At call site $q$, the QA25 circuit uses the exact encoding of $\tilde{A}\mathrm{/}d$ to simulate $e^{\pm i\tilde{A}{t}_q}$, and has a certified clean-input isometry error ${\xi }_q$ relative to that unitary; the actual controlled or inverse form has the same bound and is separately charged. Choose the value precision and simulator precisions so that ${\sum }_{q\in \mathcal{Q}}{\xi }_q+{ϵ}_{\mathrm{v}\mathrm{a}\mathrm{l}}{\sum }_{q\in \mathcal{Q}}\mathrm{\mid }{t}_q\mathrm{\mid }\le {\eta }_S\mathrm{.}$ Here ${\eta }_S$ is the total clean-subspace error for the two base lists, not an error claim on arbitrary dirty work inputs. The matrix term follows from the finite-dimensional Duhamel bound $\mathrm{\parallel }{e}^{\pm iAt}-e^{\pm i\tilde{A}t}{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le \mathrm{\mid }t\mathrm{\mid }{ϵ}_{\mathrm{v}\mathrm{a}\mathrm{l}}$; the clean-subspace unitary hybrid then sums all call-site errors. If these value, block, or simulation budgets cannot be certified, return INVALID_ACCESS before production. Their bit precision and access costs remain explicit.

Efficient inverse mode. Put $m=\lceil {\log }_2\kappa \rceil +1$ and $e_{\ast }=\min (e,\rho )$ and use this strengthened accuracy in all efficient-mode budgets below. Set $n=m+1$, ${\varphi }_j=2^{-j}$ and ${\alpha }_{\max }=2\mathrm{/}{\varphi }_m$. Use the exact QA25 sparse block encoding of $\tilde{A}\mathrm{/}d$ specified above, with every inverse/control form used and its precision/cost; its relative error to the ideal $A\mathrm{/}d$ is charged in ${ϵ}_{\mathrm{B}\mathrm{E},j}$. For each stage generate the OP1A polynomial from QSVT regularized pseudoinverse transformation with gap ${\delta }_j={\varphi }_j\mathrm{/}d$ and error $e_j=\min ({\delta }_j\mathrm{/}2,{\eta }_I\mathrm{/}8)$; this invokes its scalar generation instructions only, rather than assuming the stage input has a global gap ${\varphi }_j$. Execute the QA24 real odd QSVT circuit on the adjoint encoding, with its actual real-part-extraction control and zero-workspace success projector. Write $p_j$ for the generated OP1A polynomial. Before phase-synthesis shrink/round and implementation errors, the real odd QSVT compression on the ideal block $A\mathrm{/}d$ has eigenvalue multiplier $\mathrm{sgn}(\lambda )p_j(\mathrm{\mid }\lambda \mathrm{\mid }\mathrm{/}d)$, which approximates ${\varphi }_j\mathrm{/}(2\lambda )$ within $e_j$ for $\mathrm{\mid }\lambda \mathrm{\mid }\ge {\varphi }_j$. The actual compression additionally incurs the certified QSVT input-block, phase-synthesis and implementation errors; off that supported subspace it is simply the declared bounded circuit. Compute a success flag for the whole zero-workspace projector, and dilute its successful component by a separately supplied/constructed flag rotation of amplitude ${\varphi }_m\mathrm{/}{\varphi }_j$. The diluted target multiplier is $1\mathrm{/}({\alpha }_{\max }\lambda )$ on $\mathrm{\mid }\lambda \mathrm{\mid }\ge {\varphi }_j$, with total compression error at most ${\eta }_I$ under the combined budget below; it is an approximation even with exact block access. No unspecified PREPARE or SELECT operation occurs in this mode.

In this mode choose the least admissible dyadic QPE scales and least admissible odd repetition counts, and construct the corresponding simulator circuits at their declared precision. Set ${\eta }_I=e_{\ast }\mathrm{/}(4096m{\alpha }_{\max })$. Choose $\zeta ={\eta }_S$ as the largest dyadic number no greater than $\min (e_{\ast }\mathrm{/}(4096m{\alpha }_{\max }),1\mathrm{/}(4096m{\kappa }^2))$, so each is greater than half that minimum. In the finite call-site multiset $\mathcal{Q}$, let $Q_0=\mathrm{\mid }\mathcal{Q}\mathrm{\mid }$ and $T_0={\sum }_q\mathrm{\mid }{t}_q\mathrm{\mid }$. Request the same simulator tolerance $\xi ={\eta }_S\mathrm{/}(2\max (1,Q_0))$ at every call site and require ${ϵ}_{\mathrm{v}\mathrm{a}\mathrm{l}}{T}_0\le {\eta }_S\mathrm{/}2$ as well as the stage block-error tests. Use the least admissible simulator degrees for these requested tolerances. Smaller actual certified errors are allowed, but do not cause a smaller tolerance to be requested or additional simulator repetitions. For each stage $j$, pass phase-synthesis tolerance $e_j\mathrm{/}100$ to its QSP call and allocate the scalar approximation error $e_j$, QSVT input-block error, polynomial shrink, angle, compiler and circuit errors jointly so the actual diluted inverse compression error relative to the target multiplier is at most ${\eta }_I$. The ${ϵ}_{\mathrm{B}\mathrm{E},j}$ constraint above includes the actual generated degree $D_j$ and the finite-value/block error of $\tilde{A}\mathrm{/}d$ relative to $A\mathrm{/}d$. For a stage with $N_j\ge 1$ implemented phase/flag/projector slots, request tolerance ${\eta }_I\mathrm{/}(8N_j)$ at each such slot (including errors from phase descriptors, angles and compilation in this allocation). Sum their actual errors within this budget. These concrete tolerance choices are part of the efficient constructor; higher implementation costs supplied by the access model remain separately charged. If an OP1A polynomial/phase constructor returns a declared non-success status, propagate it before any QSVT call or success flag; the theorem's success guarantee applies when uncapped construction completes. The QA25 small-time-aware simulator is invoked only through its clean-input isometry contract, with an initially clean embedding and the controlled/inverse clean-subspace hybrid of Unitary hybrids on a preserved clean-work subspace. Within repeated calls and reversal, retain the actual work coherently; only the ideal comparison trajectory has clean work, and actual approximate work is not reset or discarded. Numerical accuracy, all signal/query/gate counts, coefficient/phase generation and arithmetic bit implementation are separate input cost coordinates. The QA23 efficient numerical arithmetic mode grants its stated ideal exact-algebraic operation count only.

Registers and stopping semantics. Put $m=\lceil {\log }_2\kappa \rceil +1$. Use clock bits $C_1,\dots ,C_m$, success bit $F$, system register $I$, separate phase workspaces $P_j$, and coefficient workspace $Q$. Once a clock bit first records a stopped branch, every later stage is controlled on all earlier clock bits being zero. All clock, outcome, and step branches are orthogonal; there are no intermediate measurements in a branch later inverted.

Finite Fourier inverse (fourier-table mode). For each requested $\varphi \in (0,1]$ and error $e_0>0$, GenerateInverse(φ,e₀) chooses rational $Y,Z>0$ by increasing integer squares until $e^{-{\varphi }^2{Y}^2\mathrm{/}2}\mathrm{/}\varphi \le e_0\mathrm{/}8,2Y{e}^{-Z^2\mathrm{/}2}\mathrm{/}\sqrt{2\pi }\le e_0\mathrm{/}8.$ Set $L_y=Z^2$, $L_z=1+Z^2+YZ$, and choose positive dyadic ${\mathrm{\Delta }}_y,{\mathrm{\Delta }}_z$ with $2YZ(L_y{\mathrm{\Delta }}_y+L_z{\mathrm{\Delta }}_z)\mathrm{/}\sqrt{2\pi }\le e_0\mathrm{/}8$. Increase $Y,Z$ to multiples of those steps and recheck. For $j=0,\dots ,J-1$, $J=Y\mathrm{/}{\mathrm{\Delta }}_y$, and $k=-K,\dots ,K-1$, $K=Z\mathrm{/}{\mathrm{\Delta }}_z$, use $y_j=j{\mathrm{\Delta }}_y$, $z_k=k{\mathrm{\Delta }}_z$, and $c_{jk}=i{\mathrm{\Delta }}_y{\mathrm{\Delta }}_z{z}_k{e}^{-z_k^2\mathrm{/}2}\mathrm{/}\sqrt{2\pi }$ for the controlled simulator $e^{-iA{y}_j{z}_k}$. Round the coefficients to exact rational complex values ${\tilde{c}}_{jk}$ with total absolute error at most $e_0\mathrm{/}8$. In lexicographic $(j,k)$ order, omit zero rounded coefficients and call the remaining list ${\beta }_{\mathrm{\ell }}$, retaining each associated simulation time. Put ${\lambda }_{\varphi }={\sum }_{\mathrm{\ell }}\mathrm{\mid }{\beta }_{\mathrm{\ell }}\mathrm{\mid }$ and choose a certified rational upper bound ${\alpha }_{\varphi }\ge \max (1,{\lambda }_{\varphi })$ for this rounded list. Execute the normalized coefficient construction below with normalization $a={\alpha }_{\varphi }$; the coefficient phases belong to SELECT, not PREPARE. If ${\tilde{c}}_{jk}$ are the used rounded coefficients and ${\xi }_{jk}$ bounds the clean-input simulation error relative to $e^{-i\tilde{A}{y}_j{z}_k}$, require $\frac{1}{{\alpha }_{\varphi }}{\sum }_{j,k}\mathrm{\mid }{\tilde{c}}_{jk}\mathrm{\mid }({\xi }_{jk}+\mathrm{\mid }{y}_j{z}_k\mathrm{\mid }{ϵ}_{\mathrm{v}\mathrm{a}\mathrm{l}})\le e_0\mathrm{/}16.$ This includes both the simulator's own precision error and the value-table error by the Duhamel bound. Allocate all other PREPARE/SELECT and controlled/inverse implementation errors jointly to at most $e_0\mathrm{/}16$, so their total normalized LCU-block error is at most $e_0\mathrm{/}8$. If these finite budgets cannot be certified, return INVALID_ACCESS (or the declared construction timeout when its cap is reached). The analytic identity being discretized is $\frac{1}{x}=\frac{i}{\sqrt{2\pi }}{\int }_0^{\mathrm{\infty }}\text{\hspace{0.17em}⁣}dy{\int }_{\mathbb{R}}z{e}^{-z^2\mathrm{/}2}{e}^{-ixyz}dz(\mathrm{\mid }x\mathrm{\mid }\ge \varphi )\mathrm{.}$ The coefficient PREPARE/SELECT block is the conditional identity proved in Projected block identity for a supplied LCU circuit. The chosen cutoffs and finite coefficients do not by themselves certify the approximation or its gate/bit cost.

Normalized coefficient construction (fourier-table mode). Given the retained rounded list and a normalization $a\ge {\lambda }_{\varphi }$, $a>0$, append two distinct identity-term labels with coefficients $+(a-{\lambda }_{\varphi })\mathrm{/}2$ and $-(a-{\lambda }_{\varphi })\mathrm{/}2$ if $a>{\lambda }_{\varphi }$; append neither if equality holds. Keep the original ordered labels followed by the positive and negative padding labels, without merging their cancelling terms. Denote this nonzero coefficient list by $({\gamma }_{\mathrm{\ell }},V_{\mathrm{\ell }}{)}_{\mathrm{\ell }<M}$, where an original $V_{\mathrm{\ell }}$ is its indicated simulation and a padding $V_{\mathrm{\ell }}$ is the identity. Even when the rounded list is empty, the two padding terms give $M\ge 1$ and total absolute coefficient sum $a$. Choose $q=\lceil {\log }_{2}M\rceil$, or a larger declared common index width, and $D=2^q$. Form the exact positive normalized table $a_{\mathrm{\ell }}=\{\begin{array}{ll}\sqrt{\mathrm{\mid }{\gamma }_{\mathrm{\ell }}\mathrm{\mid }\mathrm{/}a},& 0\le \mathrm{\ell }<M,\\ 0,& M\le \mathrm{\ell }<D\mathrm{.}\end{array}$ Use this table, whose squared amplitudes sum to one, in QC24 State Preparation, following the unit-component-scale LCU interface of LCU block-encoding construction. Its first nonzero amplitude is positive real, so its omitted reference phase is zero. Supply exact arithmetic descriptors for the amplitudes and angles in that supplier's ideal continuous-gate model; square roots need not have exact finite rational coordinate encodings. Include the actual PREPARE work in a declared clean register $W_P$ within $Q$, so $P\mathrm{\mid }{0}^q,0_{W_P}⟩={\sum }_{\mathrm{\ell }<M}{a}_{\mathrm{\ell }}\mathrm{\mid }\mathrm{\ell },0_{W_P}⟩$. Retain the complete unitary circuit, not only its zero-input state, and use its literal reversed adjoint circuit for $P^{†}$.

Supply the full SELECT unitary on all physical index labels: for $\mathrm{\ell }<M$ it applies $({\gamma }_{\mathrm{\ell }}\mathrm{/}\mathrm{\mid }{\gamma }_{\mathrm{\ell }}\mathrm{\mid })V_{\mathrm{\ell }}$ to the system and its declared simulation work, and for $M\le \mathrm{\ell }<D$ it applies identity. It acts as identity on $W_P$; all component work spaces are embedded in a common register with unused wires idle. In particular the two slack labels apply $+I$ and $-I$, respectively. Coefficient phases are not loaded into PREPARE or unprepare, where they would cancel. The returned LCU circuit is $P^{†}\mathrm{S}\mathrm{E}\mathrm{L}\mathrm{E}\mathrm{C}\mathrm{T}P$, with its success projector testing the coefficient index, $W_P$ and component work all zero. After this block, coherently XOR that all-zero predicate into the initially zero success bit $F$, using a supplied or explicit reversible predicate circuit and its charged control/work costs; this flagged circuit is the Fourier inverse call used by the stage list. Apply the conditional LCU identity to the actual component compressions with scale one; their deviations from the ideal evolutions are charged by the weighted simulation budget above, without assuming full-space simulation accuracy on dirty work. Retain all work coherently, and supply every controlled or adjoint form actually used, including the coefficient phases under an outer control; reverse the complete circuits for adjoints rather than an ideal postselected state.

Charge the complete explicit table's generation, exact descriptors, reading, storage and angle arithmetic, each PREPARE/SELECT/adjoint/control call, register and clean-work width, and precision and compilation separately. The QC24 expanded circuit uses $O((q+1)D)$ ideal bounded-arity occurrences and serial depth per generic PREPARE, with at most $\max (q-1,0)$ reusable clean work bits; it grants no finite-alphabet exactness or bit-time bound. A finite implementation instead supplies certified table/angle/whole-circuit errors, including controlled and adjoint forms, within the already allocated $e_0\mathrm{/}16$ nonsimulation LCU error budget. For example, whole-unitary PREPARE error ${\delta }_P$ and additional SELECT error ${\delta }_{\mathrm{s}\mathrm{e}\mathrm{l}}$ must include $2{\delta }_P+{\delta }_{\mathrm{s}\mathrm{e}\mathrm{l}}$ in that budget; compile complete controlled blocks so no discarded scalar becomes a relative phase. A raw rounded table without these certificates is not an implementation. This construction adds no efficient coefficient-generation, Fourier-approximation or uncapped-termination guarantee.

Common normalization (fourier-table mode). Set $L=\lceil {\log }_2(1024m\kappa \mathrm{/}e)\rceil$ and $\zeta =e\mathrm{/}{2}^L$. Generate at every ${\varphi }_j=2^{-j}$ with error $m\zeta$ and set ${\alpha }_{\max }={\max }_j{\alpha }_{{\varphi }_j}$. Rebuild each stage from its original rounded coefficient list (discarding its earlier slack labels) by the normalized coefficient construction with $a={\alpha }_{\max }$: append the explicit pair $\pm ({\alpha }_{\max }-{\lambda }_{{\varphi }_j})I\mathrm{/}2$ when its slack is positive. Use the maximum of the resulting index widths as a common $q$ and zero-pad every amplitude table to $D=2^q$; embed all stage work in the common $Q$. Rebuild PREPARE, SELECT and their required adjoint/control circuits at these sizes and recheck their actual error and cost certificates. The weighted simulation error uses denominator ${\alpha }_{\max }$ for these rebuilt blocks; the added identity terms have no simulation error. If $m{\alpha }_{\max }\zeta >e\mathrm{/}256$, increment $L$ and regenerate. Also require ${\eta }_S\le \min (e\mathrm{/}(4096m{\alpha }_{\max }),1\mathrm{/}(4096m{\kappa }^2))$ in the finite-precision simulator budget above. A declared finite cap returns CONSTRUCTION_TIMEOUT; an uncapped run relies on the separate termination and growth proof.

Gapped phase estimation and stages. Use QPE on controlled $e^{i\tilde{A}}$ with its dyadic powers constructed through the actual QA25 clean-work simulator. For each ${\varphi }_j$, GPE(φ_j,ζ) uses a dyadic phase scale at least $64\pi \mathrm{/}{\varphi }_j$, an odd number $r\ge 8\ln (1\mathrm{/}{\zeta }^2)+8$ of coherent QPE repetitions on the same eigencomponent, and reversible majority comparison of the unwrapped phase magnitude with $3{\varphi }_j\mathrm{/}2$. Threshold equality selects the large branch. The ideal comparison is QPE for $A$ with promise cases $\mathrm{\mid }\lambda \mathrm{\mid }\le {\varphi }_j$ and $\mathrm{\mid }\lambda \mathrm{\mid }\ge 2{\varphi }_j$; the transition band is unconstrained. The finite-precision implementation uses $\tilde{A}$, and its entire call-site error relative to that ideal QPE is bounded by the ${\eta }_S$ allocation above. Retain all phase work in $P_j$ and make no intermediate measurement.

At stage $j$, conditional on the active prefix $C_1=\cdots =C_{j-1}=0$, run GPE into $C_j,P_j$. On the same prefix and $C_j=1$, execute the common-normalized inverse circuit of the selected mode on $F,Q,I$; otherwise leave the branch unchanged. Let $A_j$ be this stage list, $A=A_m\cdots A_1$, and let $A_j^{\mathrm{\prime }}$ replace its inverse call by a flip of $F$, with $A^{\mathrm{\prime }}=A_m^{\mathrm{\prime }}\cdots A_1^{\mathrm{\prime }}$. The inverse lists are literal reversal with the corresponding inverse calls. The $\mathcal{Q}$-budget in the finite-precision contract includes the simulator call sites of both complete lists.

Terminal bad branch (both modes). Add a clean terminal bit $T$. After the last GPE/inverse stage, coherently set $T=1$ on the branch with all clock bits zero and leave its success flag zero. This branch is stopped and bad at the final stopping time; it is no longer unfinished. Append the same terminal-bit computation to $A^{\mathrm{\prime }}$ and reverse it inside $(A^{\mathrm{\prime }}{)}^{†}$. Thus all branches stop by $t_m$, including phase-test failures; the last stopping subspace is the orthogonal union of the ordinary last-stage stop and this terminal-bad branch. For $1\le i\le m$, define the good-or-unfinished projector as the orthogonal sum of (a) branches with $F=1$, $T=0$, and first-stop clock word $C_1=\cdots =C_{j-1}=0,C_j=1$ for some $j\le i$ (later clock bits remain zero), and (b) branches with $F=0$, $T=0$, and $C_1=\cdots =C_i=0$. Previously stopped bad branches and the terminal branch $T=1,F=0$ are excluded. Since $T$ is set after stage $m$, the unfinished term is empty at $i=m$. The final target is only $F=1$. These projectors act as identity on the clean $W_B$ factor.

Finite variable-time amplification wrapper (both modes). Pad cumulative stage counts to increasing dyadic stopping times, preserving stopped branches. Each interval budget includes the supplied input/selected reflections, controlled/inverse list implementations and reversible clock predicates; their costs remain explicit. Form the first sampler by preparing $B\mathrm{\mid }0⟩$ and running the first interval. Later $B_i$ runs the prior amplified list and the original continuation on unfinished branches only. The initial reflection is about the all-zero state before $B$; $A^{\mathrm{\prime }}$ contains the eigenvalue stages and does not undo $B$. The preparation-work factor $W_B$ stays zero throughout.

Let $n=m+1$, $c=1\mathrm{/}100$ and $p_{\min }=1\mathrm{/}(256n{\alpha }_{\max }^{})$. Use Relative estimation of a promised coherent success probability with $B_i$, the good-or-unfinished projector (the final target is $F=1$), relative error $c$, failure $\rho \mathrm{/}(16n)$ and floor $p_{\min }$. The efficient-QSVT mode selects its efficient-grid rounding branch; the Fourier-table mode retains its declared rounding branch. This is an arbitrary-sampler estimate, not a Boolean marked-count estimate. Every controlled sampler/inverse and reflection is supplied or built gatewise from an explicit circuit and charged; missing access returns INVALID_ACCESS. All estimator registers are fresh construction data, discarded before fresh production execution. A returned estimate ${\widehat{q}}_i=0$ gives ESTIMATION_FAIL. If ${\widehat{q}}_i>1\mathrm{/}(9n)$, use $V_i=B_i$. Otherwise choose the least nonnegative integer $k$ with $(2k+1{)}^2{\widehat{q}}_i\ge 1\mathrm{/}(9n)$, within $k\le \lceil 1\mathrm{/}\sqrt{p_{\min }}\rceil$; exact rational comparisons or integer binary search implement this selection. A missing crossing gives ESTIMATION_FAIL. Execute exactly $k$ ordinary iterates with this sampler and projector. The inequality $(2k+1{)}^2{\widehat{q}}_i\le 1\mathrm{/}n$ concerns the estimate; no identical actual-success upper bound is assumed.

At the final interval apply Finite amplification from a supplied success lower bound to the selected sampler with lower bound $1\mathrm{/}(18n)$ and failure $\rho \mathrm{/}4$. Copy/measure a separate success record before applying $(A^{\mathrm{\prime }}{)}^{†}$ to erase internal phase/clock/flag work on the successful branch. In efficient-qsvt mode also apply the supplied/constructed projective instrument onto zero for every work/clock/internal-flag register, including $W_B$, after this uncompute; return the system only if every tested register is zero, and set the external herald to failure otherwise. This makes the conditional returned system state pure; its work-check error/failure is budgeted with $e_{\ast }$. The instrument and its controlled-zero reflection/cost are included in $H$. Return the system and external heralding record; a failed branch returns FAIL.

For each chosen quantum resource coordinate, construct an integer $H\ge 1$ from the actual original-stage, controlled/inverse, selected/input-reflection, dilution and predicate lists, so their interval cost is at most $H{2}^{j-1}$; enlarge $H$ to include all actually needed gatewise-control depths. For gate coordinates also include the uniform upper bound for the finite estimator's QFT, efficient-grid rounding, rational medians and loop/predicate overhead over every allowed scale through its declared cap; these scales are a finite explicit function of $p_{\min },c,\rho$. Since each sampler-list cost is at least one, this common per-call polynomial-logarithmic factor absorbs that ordinary overhead without treating it as an oracle call. Pad cumulative original stage times to $t_j=H{2}^j$. Let $r$ be the actual median repetition count of the shared estimator, using its floor and confidence. In efficient-qsvt mode impose the additional total primitive-cost cap $K=\lceil 2^{30}(r\mathrm{/}c)H{\alpha }_{\max }{n}^4\rceil \mathrm{.}$ Count construction estimation, list invocations, production, inverse/control/reflection calls and final amplification against this cap and return RESOURCE_TIMEOUT before exceeding it. Each coordinate uses its actual $H$ and corresponding cap. Classical coefficient/phase construction, exact algebraic comparisons, output rounding, compiler construction and arithmetic bit work keep their own charged coordinates. The original Fourier mode retains its original declared construction/estimation caps; its distinct table and SELECT costs are not hidden by the efficient-mode quantum cap.

This is the binding OP6 constructor with its declared additional efficient inverse mode. It defines a finite capped procedure and a separate uncapped source mode; the source CKS theorem does not automatically prove the capped wrapper's failure probability, success amplitude, or resource bounds. Every sparse simulation must use the explicitly supplied reversible location permutations and the QA25 simulator access chain; QA20 XOR row/value access alone does not supply those permutations. The efficient-QSVT mode has its separate local common-scale, stopping, estimated-amplification and quantum-resource proofs. The retained Fourier-table mode keeps all of its coefficient-generation/table/SELECT costs. Both modes charge classical generation, comparison, compiler construction and arithmetic bit work separately; quantum gate efficiency does not imply a total bit-time bound.